Find the Eigenvectors/Eigenspace [[8,0,8],[0,9,0],[0,0,9]]

Problem

[[8,0,8],[0,9,0],[0,0,9]]

Solution

1. Identify the matrix A and the characteristic equation det(A−λ*I)=0 to find the eigenvalues.

A=[[8,0,8],[0,9,0],[0,0,9]]

2. Calculate the determinant of the upper triangular matrix A−λ*I which is the product of the diagonal entries.

det(8−λ)=(8−λ)*(9−λ)2

3. Solve for the eigenvalues by setting the characteristic polynomial to zero.

(λ_1)=8

(λ_2)=9

4. Find the eigenspace for (λ_1)=8 by solving (A−8*I)*v=0

[[0,0,8],[0,1,0],[0,0,1]]*[[x],[y],[z]]=[[0],[0],[0]]

⇒8*z=0⇒z=0

⇒y=0

x=free variable

(E_8)=span*{[1],[0],[0]}

5. Find the eigenspace for (λ_2)=9 by solving (A−9*I)*v=0

[[−1,0,8],[0,0,0],[0,0,0]]*[[x],[y],[z]]=[[0],[0],[0]]

⇒−x+8*z=0⇒x=8*z

y=free variable

z=free variable

[[x],[y],[z]]=[[8*z],[y],[z]]=y*[[0],[1],[0]]+z*[[8],[0],[1]]

(E_9)=span*{[[0],[1],[0]],[[8],[0],[1]]}

Final Answer

(E_8)=span*{[1],[0],[0]},(E_9)=span*{[[0],[1],[0]],[[8],[0],[1]]}

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